Universal multiple outlier hypothesis testing

The universal multiple outlier hypothesis testing problem is studied in two settings. In the first setting, each outlier can be arbitrarily distributed, and the number of outliers is fixed and known. In the second setting, the number of outliers is unknown at the outset. Nothing is known about the typical and outlier distributions other than that they are different and have full supports. For the first setting, a universally exponentially consistent test is proposed, and its achievable error exponent is characterized. The limiting error exponent achieved by such test is analyzed as the number of coordinates goes to infinity, and it is shown that the test also enjoys universally asymptotically exponential consistency. For the second setting, it is shown that with the assumption of outliers being identically distributed and the exclusion of the null hypothesis, a test based on the generalize likelihood principle is universally exponentially consistent.